On the uniqueness of algebraic curves passing through $n$-independent nodes

Abstract

A set of nodes is called $n$-independent if each its node has a fundamental polynomial of degree $n.$ We proved in a previous paper [H. Hakopian and S. Toroyan, On the minimal number of nodes determining uniquelly algebraic curves, accepted in Proceedings of YSU] that the minimal number of $n$-independent nodes determining uniquely the curve of degree $k\le n$ equals to ${\mathcal K}:=(1/2)(k-1)(2n+4-k)+2.$ Or, more precisely, for any $n$-independent set of cardinality ${\mathcal K}$ there is at most one curve of degree $k\le n$ passing through its nodes, while there are $n$-independent node sets of cardinality ${\mathcal K}-1$ through which pass at least two such curves. In this paper we bring a simple characterization of the latter sets. Namely, we prove that if two curves of degree $k\le n$ pass through the nodes of an $n$-independent node set ${\mathcal X}$ of cardinality ${\mathcal K}-1$ then all the nodes of ${\mathcal X}$ but one belong to a (maximal) curve of degree $k-1.$